Lemma 2.2.3

Let be a projection in an algebra , and let be a self-adjoint element in . Put then Proof:
Since is self-adjoint, its spectrum consists of real numbers. Recall that . It suffices to show that if is such that then .
For such a real number , the element is invertible in and Not exactly sure how to get this other than via translation
It follows that

It follows that is invertible (This is not hard to see since the spectrum of p is either 0 or 1, so the term on the left ought to be invertible, so just multiply on the left by its inverse and by group closure we have what we need. ), and so this implies that is invertible, which shows that